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NELL-1 is a secreted, osteoinductive protein whose expression rheostatically controls skeletal ossification. Overexpression of NELL-1 results in craniosynostosis in humans and mice, whereas lack of Nell-1 expression is associated with skeletal undermineralization. Here we show that Nell-1 -haploinsufficient mice have normal skeletal development but undergo age-related osteoporosis, characterized by a reduction in osteoblast:osteoclast (OB:OC) ratio and increased bone fragility. Recombinant NELL-1 binds to integrin β1 and consequently induces Wnt/β-catenin signalling, associated with increased OB differentiation and inhibition of OC-directed bone resorption. Systemic delivery of NELL-1 to mice with gonadectomy-induced osteoporosis results in improved bone mineral density. When extended to a large animal model, local delivery of NELL-1 to osteoporotic sheep spine leads to significant increase in bone formation. Altogether, these findings suggest that NELL-1 deficiency plays a role in osteoporosis and demonstrate the potential utility of NELL-1 as a combination anabolic/antiosteoclastic therapeutic for bone loss. The growth factor NELL-1 induces bone formation during development, but its role in osteoporosis is unknown. This study shows that NELL-1 binding to integrin ß1 induces Wnt/ß-catenin signalling in the bone and restores bone mineral density in osteoporotic mice and sheep, suggesting the therapeutic potential of NELL-1 for the treatment of bone loss.

A Correction to this paper has been published: https://doi.org/10.1038/s41467-021-20933-x .

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory (see [S. F. Du, D. Marusic, and A. O. Waller, J. Combin. Theory Ser. B, 74 (1998), pp. 276--290], [S. F. Du, J. H. Kwak, and M. Y. Xu, J. Combin. Theory Ser. B, 93 (2005), pp. 73--93], [R. Feng, J. H. Kwak, J. Kim, and J. Lee, SIAM J. Discrete Math., 11 (1998), pp. 265--272], [R. Feng. and J. H. Kwak, Discrete Math., 277 (2004), pp. 73--85], [C. D. Godsil and A. D. Hensel, J. Combin. Theory Ser. B., 56 (1992), pp. 205--238], [M. Hofmeister, Discrete Math., 143 (1995), pp. 87--97], [M. Hofmeister, SIAM J. Discrete Math., 8 (1995), pp. 51--61], [M. Hofmeister, SIAM J. Discrete Math., 11 (1998), pp. 286--292], [J. H. Kwak, J. Chun, and J. Lee, SIAM J. Discrete Math., 11 (1998), pp. 273--285], [J. H. Kwak and J. Lee, Canad. J. Math., 42 (1990), pp. 747--761], and [J. H. Kwak and J. Lee, Combinatorial and Computational Mathematics: Present and Future, (2001), pp. 97--161]). A covering projection is called circulant if its covering graph is circulant. A covering projection p from a Cayley graph${\\rm Cay} ({\\cal A},X)$onto another${\\rm Cay} ({\\cal Q},Y)$is called typical if the map$p: {\\cal A}\\rightarrow {\\cal Q}$on the vertex sets is a group homomorphism from${\\cal A}$onto${\\cal Q}$ . In [R. Feng. and J. H. Kwak, Discrete Math., 277 (2004), pp. 73--85], the authors enumerated the isomorphism classes of typical circulant double covering projections onto a circulant graph. As a continuation of this work, we enumerate in this paper the isomorphism classes of those covering projections of any folding number.

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. An enumeration of the isomorphism classes of n-fold coverings of a graph G was done by Kwak and Lee [Canad. J. Math., XLII (1990), pp. 747--761] and independently by Hofmeister [Discrete Math., 98 (1991), pp. 437--444]. An enumeration of the isomorphism classes of connected n-fold coverings of a graph G was recently done by Kwak and Lee [J. Graph Theory, 23 (1996), pp. 105--109]. But the enumeration of the isomorphism classes of regular coverings of a graph G has been done for only a few cases. In fact, the isomorphism classes of${\\cal A}$ -coverings of G were enumerated when${\\cal A}$is the cyclic group$\\BZ_n$ , the dihedral group$\\BD_n$(n: odd), and the direct sum of$m$copies of$\\BZ_p$ . (See [Discrete Math., 143 (1995), pp. 87--97], [J. Graph Theory, 15 (1993), pp. 621--627], and [Discrete Math., 148 (1996), pp. 85--105]). In this paper, we discuss a method to enumerate the isomorphism classes of connected${\\cal A}$ -coverings of a graph G for any finite group${\\cal A}$and derive some formulas for enumerating the isomorphism classes of regular n-fold coverings for any natural number n. In particular, we calculate the number of the isomorphism classes of${\\cal A}$ -coverings of G when${\\cal A}$is a finite abelian group or the dihedral group$\\BD_n$ . Our method gives partial answers to the open problems 1 and 2 in [Discrete Math., 148 (1996), pp. 85--105] and also gives a formula to calculate the number of the subgroups of a given index of any finitely generated free abelian group.

Hofmeister introduced the notion of a concrete (resp., concrete regular) covering of a graph G and gave formulas for enumerating the isomorphism classes of concrete (resp., concrete regular) coverings of G [Ars Combin., 32 (1991), pp. 121--127; SIAM J. Discrete Math., 8 (1995), pp. 51--61]. In this paper, we show that the number of the isomorphism classes of n-fold concrete (resp., concrete regular) coverings of G is equal to that of the isomorphism classes of n-fold (resp., regular) coverings of a new graph, the join$G+\\infty$of G and an extra vertex$\\infty$ . As a consequence, we can enumerate the isomorphism classes of concrete (resp., concrete regular) coverings of a graph by using known formulas for enumerating the isomorphism classes of coverings (resp., regular coverings) of a graph.

Lots of graphs having a symmetry property can be described as coverings of simpler graphs. In this manuscript, we examine several enumeration problems for various types of nonisomorphic graph coverings of a graph and some of their applications to a group theory or to a surface theory. This manuscript is organized as follows. In section 1, we introduce basic concepts. In section 2, by using covering graph construction, we count the positive isomorphism classes of cycle permutation graphs, which is equal to the number of double cosets of the dihedral group 𝔻n in the symmetric group Sn on n elements. In section 3, we count nonisomorphic (connected) coverings of a graph and, as its application, we have another recursive formula for the number of conjugacy classes of subgroups of given index of a finitely generated free group. In section 4, we count nonisomorphic regular coverings of a graph whose covering transformation groups are abelian and, as its application, we count subgroups of given index of free abelian groups. The same work is done in section 5 for regular coverings having dihedral voltage groups. In section 6, we discuss a general counting formula for regular coverings having any finite voltage group. In section 7, after discussing a combinatorial proof of Hurwitz theorem for surface branched coverings, we consider the number of subgroups of surface groups. Finally, in section 8, we discuss a distribution of branched surface coverings of surfaces and some related topological properties including a generalization of the classical Alexander theorem.

The main purpose of this thesis is to define a free cyclic group action of prime order on a homotopy sphere which bounds a parallelizable manifold in a concrete way, and to show an algebraic characterization of its orbit space, called a lens space, to be stably parallelizable. Some special cases of these lens spaces have been considered by P. Orlik who defined a cyclic group action on a homotopy sphere represented as a Brieskorn manifold. The author also described the tangent bundle of a lens space and computed the characteristic classes of the lens space, correcting some errors which occurred in P. Orlik's paper (Smooth homotopy lens spaces, Michigan Math. J., 16 (1969) 245-255). By using the algebraic characterization of a lens space to be stably parallelizable, it is shown that \"there exists a stably parallelizable (2n-1)-dimensional lens space if and only if n is less than the order of the cyclic group which defines the lens space\". This gives some interesting results. For example, there are many ways to define free Z(,5)-actions on S('5) which are not linear so that these lens spaces are not diffeomorphic to any 5-dimensional classical lens space L(5; q(,1), q(,2), q(,3)). On 7-dimensional homotopy spheres, the author showed the following result: For p (GREATERTHEQ) 5 prime, there is a Z(,p)-action on C('5) and representative (SIGMA)'s of all 28 diffeomorphism classes of 7-dimensional homotopy spheres as submanifolds of C('5) such that the restriction of the Z(,5)-action on each homotopy sphere (SIGMA) is well-defined, and all induced lens spaces (SIGMA)/X(,p) are stably parallelizable\". Furthermore, for n (GREATERTHEQ) 3 odd, p (GREATERTHEQ) n, the standard sphere or the Kervaire sphere of dimension 2n-1 admits a free Z(,p)-action so that its orbit space is stably parallelizable. Finally, the author showed two generations of his work: one comes from a generalized description of a homotopy sphere as an algebraic variety, the other deals with the case of the composite moduli.

Isomerases perform biotransformations without cofactors but often cause an undesirable mixture of substrate and product due to unfavorable thermodynamic equilibria. We demonstrate the feasibility of using an engineered yeast strain harboring oxidoreductase reactions to overcome the thermodynamic limit of an isomerization reaction. Specifically, a yeast strain capable of consuming lactose intracellularly is engineered to produce tagatose from lactose through three layers of manipulations. First, GAL1 coding for galactose kinase is deleted to eliminate galactose utilization. Second, heterologous xylose reductase (XR) and galactitol dehydrogenase (GDH) are introduced into the ∆gal1 strain. Third, the expression levels of XR and GDH are adjusted to maximize tagatose production. The resulting engineered yeast produces 37.69 g/L of tagatose from lactose with a tagatose and galactose ratio of 9:1 in the reaction broth. These results suggest that in vivo oxidoreaductase reactions can be employed to replace isomerases in vitro for biotransformation. A desired product cannot be obtained at higher concentration than its equilibrium concentration when isomerases are used for biotransformation. Here, the authors engineer in vivo oxidoreductive reactions in yeast to overcome the equilibrium limitation of in vitro isomerases-based tagatose production.

There are limited therapeutic options for extensively drug-resistant tuberculosis. In this study from South Korea, linezolid was shown to have some activity in treating resistant tuberculosis; however, its use was associated with clinically significant toxicity. Linezolid (Zyvox, Pfizer) was approved in 2000 for drug-resistant, gram-positive bacterial infections. 1 A member of the oxazolidinone antibiotic class, linezolid inhibits protein synthesis by binding the 23S ribosomal RNA (rRNA) portion of the bacterial 50S ribosomal subunit. 2 In adults, linezolid is administered at a dose of 600 mg twice daily, with phase 3 and postmarketing trials showing an acceptable side-effect and adverse-event profile during the FDA-approved 28 days of therapy. 3 Data on longer-term use are limited, but serious neuropathies (e.g., peripheral and optic neuropathies), myelosuppression, and hyperlactatemia have been observed 4 , 5 and are considered to be related to the inhibition . . .